On the Role of Information Theoretic Uncertainty Relations in Quantum Theory

TL;DR

This paper develops new information-theoretic uncertainty relations using Rényi entropy, demonstrating their superiority over traditional relations in quantum models like two-level systems and complex states.

Contribution

It introduces a novel class of uncertainty relations based on Rényi entropy, extending previous information-theoretic approaches and providing improved bounds in quantum mechanics.

Findings

New uncertainty relations outperform traditional ones in quantum models.

Enhanced bounds for heavy-tailed wave functions and Schrödinger cat states.

Discussion of generalized entropy power inequalities and geometric interpretations.

Abstract

Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) R\'{e}nyi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual Robertson-Schr\"{o}dinger uncertainty relation and Kraus-Maassen Shannon entropy based uncertainty relation. In the continuous case the ensuing entropy power uncertainty relations are discussed in the context of heavy tailed wave functions and Schr\"odinger cat states. Again, improvement over both the Robertson-Schr\"{o}dinger uncertainty principle and Shannon ITUR is demonstrated in these cases. Further salient issues such as the proof of a generalized entropy power inequality and a geometric picture of information-theoretic uncertainty relations are also discussed.